Question: Solve for $x$ : $ 2|x - 5| + 3 = 4|x - 5| + 4 $
Explanation: Subtract $ {2|x - 5|} $ from both sides: $ \begin{eqnarray} 2|x - 5| + 3 &=& 4|x - 5| + 4 \\ \\ {- 2|x - 5|} && {- 2|x - 5|} \\ \\ 3 &=& 2|x - 5| + 4 \end{eqnarray} $ Subtract $4$ from both sides: $ \begin{eqnarray} 3 &=& 2|x - 5| + 4 \\ \\ {- 4} && {- 4} \\ \\ -1 &=& 2|x - 5| \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{-1} {{2}} = \dfrac{2|x - 5|} {{2}} $ Simplify: $ -\dfrac{1}{2} = |x - 5| $ The absolute value cannot be negative. Therefore, there is no solution.